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Climate Generator(Stochastic Climate Representation: 120 ka to
present year)
by
c© Mohammad Hizbul Bahar ArifMaster of Science
A thesis submitted to theSchool of Graduate Studiesin partial
fulfillment of the
requirements for the degree ofMaster of Science.
Scientific Computing ProgramMemorial University of
Newfoundland
16th December 2016
St. John’s Newfoundland
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Contents
Abstract 4
Acknowledgements 5
Abbreviations 6
List of Figures 11
1 Introduction 12
1.1 Research Context . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 12
1.2 Downscaling . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 14
1.3 Bayesian inference . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 16
1.4 Bayesian Artificial Neural Network (BANN) . . . . . . . . .
. . . . . 16
1.5 Markov Chain Monte Carlo integration (MCMC) . . . . . . . .
. . . 18
1.6 Gibbs sampling . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 18
1.7 Network architecture and prior computation . . . . . . . . .
. . . . . 18
1.8 Automatic Relevance Determination (ARD) . . . . . . . . . .
. . . . 19
1.9 Climate Turing Test . . . . . . . . . . . . . . . . . . . .
. . . . . . . 20
1.10 Climate Generator (CG) . . . . . . . . . . . . . . . . . .
. . . . . . . 20
1.10.1 Reasoning behind the name of CG . . . . . . . . . . . . .
. . 20
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1.10.2 CG predictors . . . . . . . . . . . . . . . . . . . . . .
. . . . . 21
1.10.3 Climate Generator and reality . . . . . . . . . . . . . .
. . . . 22
1.10.4 Present setup and future prospect . . . . . . . . . . . .
. . . . 22
1.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 23
1.12 Thesis Overview . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 23
2 Stochastic climate representation for millennial scale
integration over
North America 24
2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 24
2.2 keywords . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 25
2.3 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 25
2.4 Test study region and data . . . . . . . . . . . . . . . . .
. . . . . . . 26
2.5 Methods . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 27
2.5.1 BANN design and training . . . . . . . . . . . . . . . . .
. . . 28
2.5.2 BANN Implementation . . . . . . . . . . . . . . . . . . .
. . . 29
2.5.3 Adding noise . . . . . . . . . . . . . . . . . . . . . . .
. . . . 31
2.6 Implementation Results . . . . . . . . . . . . . . . . . . .
. . . . . . 31
2.6.1 Selection of BANN architecture . . . . . . . . . . . . . .
. . . 32
2.6.2 Model Comparison . . . . . . . . . . . . . . . . . . . . .
. . . 38
2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 45
2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 46
2.9 Co-authorship statement . . . . . . . . . . . . . . . . . .
. . . . . . . 47
A Supplement of Climate Generator 51
3
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Abstract
I present a computationally efficient stochastic climate model
for large spatiotemporal
scales (example, for the context of glacial cycle modelling). In
analogy with a Weather
Generator (WG), the model can be thought of as a Climate
Generator (CG). The
CG produces a synthetic climatology conditioned on various
inputs. Inputs for the
CG include the monthly mean sea surface temperature field from a
simplified Energy
Balance Model (EBM), surface elevation, surface ice, carbon
dioxide, methane, orbital
forcing, latitude and longitude. The CG outputs mean monthly
surface temperature
and precipitation using Bayesian Artificial Neural Networks
(BANN) for non-linear
regression. The CG is trained against the results of GCMs
(FAMOUS and CCSM)
over the last deglacial (22 ka to present). For validation, CG
predictions are compared
directly against the 120 ka to 22.05 ka interval of FAMOUS
results that were not
used for CG training. The stochastic noise is added to each
prediction by generating
the random normal distribution with mean from the ensemble
networks for a single
guess and Standard deviation computed from 10th and 90th
percentile of the BANN
predictive distribution for each time step. For the CG trained
against FAMOUS, I
show the predictive errors (relative to FAMOUS) are comparable
to the difference
between FAMOUS and the CCSM.
4
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Acknowledgements
I would offer deep gratitude to my supervisor Lev Tarasov
(Associate professor,
Department of Physics and physical oceanography) for his
guidance, inspiration and
valuable suggestions throughout this research. Financial support
was provided by MUN
Scientific Computation program and RDC. Without this support, it
would not have
been possible to pursue this research on a full-time basis. This
work has benefited
greatly from the instruction provided by Tristan Hauser. Also,
by assistance discus-
sions with Taimaz Bahadory, Kevin Le Morzadec, Ryan Love and
Benoit Lecavalier.
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Abbreviations
ANN Artificial Neural Network
BANN Bayesian Artificial Neural Network
CCSM Community Climate System Model
WG Weather Generator
CG Climate Generator
CCT Climate Turing Test
CGfamous CG train with FAMOUS climate model data
CGccsm CG train with CCSM climate model data
EMIC Earth systems Model of Intermediate Complexity
EOFs Empirical Orthogonal Functions
GCM General Circulation Model
ka kilo year before present
MCMC Markov Chain Monte Carlo
RMSE Root Mean Square Error
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List of Tables
2.1 Predictor sets for displayed CG results . . . . . . . . . .
. . . . . . . . . . . . . . . . 32
2.2 Different models based on architecture and inputs . . . . .
. . . . . . . . . . . . . . 33
2.3 MEAN DIFFERENCE (MD) and RMSE relative to the FAMOUS (except
for last
entry) over the full grid area (train period) . . . . . . . . .
. . . . . . . . . . . . . . 33
2.4 MEAN DIFFERENCE (MD) and RMSE relative to the FAMOUS (Test
period) over
the full grid area . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 34
2.5 ARD analysis for the Model F (CGfamous) . . . . . . . . . .
. . . . . . . . . . . . . 37
2.6 Mean Difference (MD), Correlation (COR) and RMSE relative to
FAMOUS (except
for last entry) over the ice region (Train period) . . . . . . .
. . . . . . . . . . . . . . 39
2.7 Mean Difference (MD) , Correlation (COR) and RMSE relative
to FAMOUS (Test
period) over the ice region . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 39
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List of Figures
2.1 Final ANN Diagram (Architecture type I) used in CGfamous and
CGccsm prediction 29
2.2 ANN Diagrams are tested to improve CG prediction . . . . . .
. . . . . . . . . . . . 32
2.3 Taylor diagram displaying a statistical comparison with
FAMOUS (test part) of six
different CG estimates. Pattern similarities are quantified
based on their correlation
and centered root mean square difference between CG and FAMOUS,
and standard
deviation with respect to the mean of the corresponding field.
Contour grey lines
indicate the root mean square (RMS) values. The model M is the
same as model F
but trained with CCSM climate model data. . . . . . . . . . . .
. . . . . . . . . . . . 35
2.4 Taylor diagram displaying a statistical comparison with
FAMOUS (test part) of six
different CG estimates. Pattern similarities are quantified
based on their correlation
and centered root mean square difference between CG and FAMOUS,
and standard
deviation with respect to the mean of the corresponding field.
Contours grey line
indicates the root mean square (RMS) values. The model M is the
same as model F
but trained with CCSM climate model data. . . . . . . . . . . .
. . . . . . . . . . . . 36
2.5 Comparison of the spatial mean (with latitudinal weighting)
temperature time series.
The black vertical line separates the test (left) and training
part (right). . . . . . . . 40
2.6 The August temperature field (Deg C) at 100 ka (1st and 2nd
row) with elevation
and ice contours shown in black and blue. The difference between
plots are shown in
the 3rd row. Model names are indicated in the top left corner in
each box. . . . . . . 41
2.7 February precipitation field (cm/month). Left column: test
case. Right column:
Training Case. Model names and times are indicated in the top
left corner in each box. 42
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2.8 Temperature field, left column: Distribution of the
expansion coefficient over total
time steps for the leading two EOFs of the FAMOUS Model data
(black), and the
distribution of the expansion coefficients over time obtained by
an ensemble projection
of CGfamous simulation (red), EBM temperature (blue) and CCSM
(cyan) onto the
same EOFs. Top time series represent the August EOFs and bottom
time series
are the February EOFs. Right column: Distribution of the
expansion coefficients for
the leading two FAMOUS EOFs for the FAMOUS (black), CGfamous
(red), EBM
temperature (blue), and CCSM (cyan) datasets . . . . . . . . . .
. . . . . . . . . . . 43
2.9 Precipitation field, left column: Distribution of the
expansion coefficient over total
time steps for the leading two EOFs of the FAMOUS Model data
(black), and the
distribution of the expansion coefficients over time obtained by
an ensemble projection
of CGfamous simulation (red) and CCSM (cyan) onto the same EOFs.
Top time series
represent the August EOFs and bottom time series are the
February EOFs. Right
column: Distribution of the expansion coefficients for the
leading two FAMOUS EOFs
for the FAMOUS (black), CGfamous (red) and CCSM (cyan) datasets
. . . . . . . . 44
A.1 Comparison of the spatial variance (standard deviation) on
the ice region. The black
vertical line separates test (left) and training part (right). .
. . . . . . . . . . . . . . 52
A.2 Comparison of the spatial mean (with latitudinal weighting)
precipitation time series.
The black vertical line separates test (left) and training part
(right). . . . . . . . . . 53
A.3 February temperature field (Deg C) at 18 ka with the
elevation and ice contour shown
in black and blue. Difference between plots are shown in 3rd
row. Model names and
months are indicated in the top left corner in each box). . . .
. . . . . . . . . . . . . 54
A.4 August temperature field (Deg C) at 18 ka with the elevation
and ice contour shown
in black and blue. Difference between plots are shown in 3rd
row. Model names and
months are indicated in the top left corner in each box). . . .
. . . . . . . . . . . . . 55
A.5 February precipitation field (cm/month) at 18 ka with the
elevation and ice contour
shown in black and blue. Models name and month are indicated in
the top left corner
in each box. Difference between plots are shown in 3rd row. . .
. . . . . . . . . . . . 56
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A.6 August precipitation field (cm/month) at 18 ka with the
elevation and ice contour
shown in black and blue. Difference between plots are shown in
3rd row. Model
names and month are indicated in the top left corner in each
box). . . . . . . . . . . 57
A.7 The August temperature field (Deg C). (1st and 2nd row) with
the elevation and ice
contour shown in black and blue. Difference between plots are
shown in 3rd row.
Model names are indicated in the top left corner in each box). .
. . . . . . . . . . . . 58
A.8 The February temperature field (Deg C) at 100 ka (1st and
2nd row) with the elevation
and ice contour shown in black and blue respectively. Difference
between plots are
shown in 3rd row. Model names are indicated in the top left
corner in each box. . . 59
A.9 The August temperature field (Deg C) at 80 ka (1st and 2nd
row) with the elevation
and ice contour shown in black and blue respectively. Difference
between plots are
shown in 3rd row. Model names are indicated in the top left
corner in each box. . . 60
A.10 The February temperature field (Deg C) at 80 ka (1st and
2nd row) with the elevation
and ice contour shown in black and blue respectively. Difference
between plots are
shown in 3rd row. Model names are indicated in the top left
corner in each box. . . 61
A.11 The August temperature field at 60 ka (Deg C) (1st and 2nd
row) with the elevation
and ice contour shown in black and blue respectively. Difference
between plots are
shown in 3rd row. Model names are indicated in the top left
corner in each box. . . 62
A.12 The February temperature field (Deg C) at 60 ka (1st and
2nd row) with the elevation
and ice contour shown in black and blue respectively. Difference
between plots are
shown in 3rd row. Model names are indicated in the top left
corner in each box. . . 63
A.13 February precipitation field (cm/month). Models name and
times indicated in the
top left corner in each box. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 64
A.14 August precipitation field (cm/month) (1st and 2nd row).
Difference between plots
are shown in 3rd row. Model names and times are indicated in the
top left corner in
each box. . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 65
A.15 The February precipitation field (cm/month). Model names
and times are indicated
in the top left corner in each box. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 66
A.16 The August precipitation field (cm/month). Model names and
times are indicated in
the top left corner in each box. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 67
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A.17 February correlation map between residual (i.e. CG without
noise - FAMOUS) Tem-
perature and precipitation fields. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 68
A.18 August correlation map between residual (i.e. CG without
noise - FAMOUS) Tem-
perature and precipitation fields. . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 69
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Chapter 1
Introduction
1.1 Research Context
Computational cost is a key issue for glacial cycle modelling,
particularly for paleoclimate modelling.
For large spatio-temporal time-scales even previous generation
GCMs are prohibitively expensive,
involving millions of cpu-hours for a single simulation. The
statistical correction of faster simplified
climate models to better approximate the predictive quality of
GCMs constitutes the central theme
of this thesis. From a Bayesian framework, I extract a posterior
for climate conditioned on various
inputs, including the output of a fast 2D Energy Balance Model
(EBM). Natural variability or climate
noise are imposed in model prediction to each time steps through
the addition of gaussian noise,
with noise variance extracted from the afore mentioned
posterior. I focus on constructing an efficient
stochastic climate representation to provide a better climate
representation for glacial cycle (120 ka
to present year) modelling. Towards this goal, the small scale
Weather Generator (WG) concept
is implemented on a large spatiotemporal scale, and accordingly,
is named a “Climate Generator
(CG)”.
General Circulation Models (GCMs) are an established tool for
estimating the large scale evolution
of the Earth’s climate. They represent the physical processes
occurring in the atmosphere, ocean,
cryosphere and land surface and their interactions. In GCMs,
core mathematical equations that
are derived from mathematical laws (conservation of energy,
conservation of mass, conservation of
momentum and the ideal gas law) are solved numerically. These
models produce a three-dimensional
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picture of the time evolution of the state of the whole climate
system. Current GCMs are too com-
putationally expensive to run continuously over O(100 kyr)
glacial cycle time scales. For example,
the simplified low resolution (atmospheric part of the model has
resolution 5 ◦ × 7.5 ◦) FAMOUS
GCM has been run for the entire last glacial cycle (LGC period,
120 ka to present year) only in an
highly accelerated mode. This model can run at the rate of 250
years in a day on eight cores [Smith
and Gregory, 2012]. The longest integration of a full complexity
GCM to date took 2 years to
complete the 22 ka to present deglacial interval (Community
Climate System Model (CCSM), with
a T31 atmospheric component [Liu et al., 2009]). As a fast
alternative, Energy Balance Models
(EBMs) can integrate a whole glacial cycle in a day or less.
They predict the surface temperature as
a function of the Earth’s energy balance with diffusive
horizontal heat transports. However, in an
EBM, atmospheric dynamics are not modelled and only the sea
level temperature field is computed
on the basis of energy conservation. Given this, the resolution
of the EBM is kept low (T11 @ 1500
km) and has no precipitation field [Hyde et al., 1990], [Tarasov
and Peltier, 1997].
Coupled icesheet and climate modelling over a full glacial cycle
is an example where computational
speeds currently preclude the use of GCM climate
representations, especially for large ensemble-
based analyses required for assessing dynamical and/or
reconstructive uncertainties. Earth System
Models of Intermediate Complexity (EMICSs) enable long-term
climate simulations over several
thousands of years but are at the edge of applicability for a
full glacial cycle. For example, LOVE-
CLIM is a low resolution (Atmospheric component is T21) climate
model. It takes about 15 days
to run 10 kyr, [Goosse et al., 2010]). Thus, there remains a
need for a faster climate representation
(temperature, precipitation etc.) for last glacial cycle (120 ka
to present year) ice sheet modelling,
especially in large ensemble contexts.
To do this, a new approach is proposed for efficient climate
modelling over large spatio-temporal
scales: the Climate Generator (CG). The CG uses the results of
previous GCM runs to effectively
improve the output of a fast simplified climate model (in this
case an EBM) and thereby provide a
stochastic representation of climate that runs approximately at
the speed of the fast model. The Cli-
mate Generator (CG) can also be understood as a field-specific
emulator for GCMs. This is because
we train our CG using GCM data to make climate predictions
without the computational expense
of running a full GCM. As an alternative view, the CG operates
similar to aspects of downscaling
tools. Downscalling tools are generally used to increase
resolution in certain climate characterist-
ics. Similarly, the CG is developed based on mainly coarse
resolution climate representations and
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converts EBM temperature to a GCM scale. The CG produces
temperature and precipitation fields
using a EBM temperature field as input, similar to downscaling
techniques for temperature.
1.2 Downscaling
Most GCMs neither assimilate nor provide information on scales
smaller than a few hundred kilo-
meters for the atmosphere. The relevant scale of climatic
impacts is much smaller than this. We
have, therefore, a spatiotemporal downscaling problem (Fowler
and Wilby [2007]). Downscaling
methods create a relationship between the state of some
variables on a large scale and the state of
the variables on a smaller scale. These methods are used to
convert the GCM output to a local
scale. The two main approaches to downscaling are Dynamical
Downscaling (DD) and Statistical
Downscaling (SD). Both methods can provide a high-resolution
regional climate, but DD is compu-
tationally expensive and requires large amounts of GCM boundary
forcing data Fowler and Wilby
[2007]. SD needs a long and reliable observed historical data
series for calibration and depends on the
choice of predictors, domain size, and climate region. But SD
approaches are simple to implement
and computationally inexpensive.
To create such a SD tool, the following conditions must be
fulfilled (Schoof [2013], [Benestad et al.,
2008]):
• A strong relationship between large-scale predictor and
small-scale predictand.
• Predictors are simulated well by the models.
• Statistical relationship between the predictor and predictand
that does not change over time.
SD methods have been categorized (based on application
technique) as regression-based methods
(Multivariate Regression (MVR), Singular Value Decomposition
(SVD), Canonical Correlation Ana-
lysis (CCA), Artificial Neural Network (ANN)), weather pattern
based method (fuzzy classification,
self-organizing map, Monte Carlo methods) and weather generators
(Markov chains, stochastic mod-
els, Schoof [2013]). Regression-based methods are relatively
straightforward to apply and simple to
handle but have an inadequate representation of observed
variance and extreme events [Wilby et al.,
2004]. CCA finds spatially coherent patterns in various data
fields that have the largest possible cor-
relation, whereas SVD finds coupled spatial patterns that have
maximum temporal covariance [Be-
nestad et al., 2008]. MVR optimizes the fit (minimizing the
RMSE). Regression-based methods
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are widely used in hydrological response assessment [Chu et al.,
2010]. Weather-pattern based
methods are often used in the analysis of extreme events.
Weather Generators (WGs) replicate the
statistical attributes of local climate variables rather than
the observed sequences of events [Wilby
et al., 2004]. Regression and weather pattern based methods have
been jointly implemented via
ANNs (e.g. combined principle components of multiple
circulations as predictors in an ANN for
winter precipitation) [Schoof, 2013]. The statistical
downscaling method (SDSM) is a hybrid of a
regression method and weather generator [Chu et al., 2010].
Statistical methods are chosen based
on the nature of local predicted variables. A relatively smooth
variable, such as monthly mean
temperature, can be reasonably represented by regression-based
methods. If the local variable is
highly discontinuous in space and time, such as daily
precipitation, it will require a more complex
non-linear approach [Benestad et al., 2008].
Artificial Neural Networks (ANNs) are simple to implement and
are non-parametric. As a result,
ANNs are widely used for modelling the complex relationship
between inputs and outputs in a
short spatial or temporal scale climate prediction, such as
forecasting problems [Reusch and Alley,
2004]. If the input and output relationships are nonlinear or if
the output is non-Gaussian, ANNs
give better fits to observations than standard parametric
approaches e.g. for the case of daily
precipitation [Schoof and Pryor, 2001], [Dibike and Coulibaly,
2006]. However, ANNs do not
provide uncertainty estimates, and over-fitting is the most
common cause of poor predictive ability.
Bayesian Artificial Neural Networks (BANNs) address these
deficiencies. BANN is a non-linear
regression-based method where Bayesian inference is implemented
on ANNs. BANN have been
used to create weather generators [Hauser and Demirov, 2013],
perform climate model calibration
[Hauser et al., 2012] and make short scale climate predictions
[Lu and Qin, 2014]. I have chosen
BANNs as an emulator to construct my CG for the following
reasons:
• BANNs are ensembles of ANNs drawn from a probability
distribution derived by training the
network against available data using Bayesian inference.
• The training procedure of BANNs estimates network parameters
based on the fitted para-
metric noise model.
• BANNs provide a prediction consisting of an expected value
together with an associated
uncertainty Lee [2007]).
• Over-fitting is generally avoided in BANNs because their
learning procedure is based on
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minimizing the sum of the bias and the variance squared.
• There is no need for separate cross validation on data sets,
so that the whole data set can be
used for training and validation.
• The amount of training data is irrelevant in BANNs for
adjusting the complexity of the
model because, in the Bayesian perspective, a posterior
distribution is not going to maximize
fit (unless the prior is uniform). In other terms, the prior
regularizes the fitting.
• Automatic Relevance Determination (ARD) can identify which
inputs are most relevant to
predict the target. This is important because including inputs
of low statistical relevance will
likely degrade predictive performance Neal [2012].
The CG uses a BANN for non linear regression.
1.3 Bayesian inference
In Bayesian inference, Bayes theorem is applied to derive the
posterior probability, θ, given the data,
D,
P (θ/D) =P (D/θ)P (θ)
P (D)∝ L(θ|D)P (θ), (1.1)
where P(θ) is the prior distribution for the parameters, (that
express our initial beliefs about their
values, before any data has arrived) and P(D/θ) is called the
likelihood. A prediction of an unknown
quantity C is given by the expection of the C relative to the
posterior distribution of the parameters,
sic
P (C|D) =∫P (C|θ)P (θ|D)dθ. (1.2)
Bayesian inference implemented on ANNs is known as Bayesian
Artificial Neural Networks (BANNs).
The BANNs procedure is briefly described in Neal [2012]. The
following sections 1.4, 1.5 and 1.7
are presented a summary of the BANNs procedure based on Neal
[2012].
1.4 Bayesian Artificial Neural Network (BANN)
In BANNs, Bayesian inference is applied to ANN to generate a
posterior probability distribution for
the network weights, W , given the training data, D. In an
ordinary ANN (with one hidden layer),
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the output is computed as follows:
fk(x) = bk +∑j
vjkhj(x)hj(x) = tanh(aj +∑i
uijxi), (1.3)
where i is the input unit, j is the hidden unit and k is the
output unit. The uij and vjk are the
weights on the connection between i to j and j to k
respectively. Biases of the hidden and output
units are given by aj and bk. These weights and biases are known
as the network parameters. Each
fk(x) (output value of the network) is weighted by the sum of
hidden values and a bias. The value of
each hidden unit hj(x), is computed with a non-linear activation
function, in our case the hyperbolic
tangent (tanh). In the case of a regression model (real value
targets), we have for distinct outputs
given the input:
P (y|x) =∏k
1√2φσk
exp(−(fk(x)− yk)2
2σ2k). (1.4)
Where the targets yk and input x are in this case modelled by
Gaussian distribution with mean
fk(x) (corresponding network outputs) and a standard deviation
given by the hyperparameters σk
(noise levels for the targets).
Consider the set of training cases, (x1, y1), (x2, y2), ...,
(xn, yn) and replacing: C = xn+1 (predictive
distribution) and D = (x1, y1), (x2, y2), ..., (xn, yn); the
equation 1.2 is modified as follows (for the
target values in new test case):
P (yn+1|xn+1, (x1, y1), (x2, y2), ..., (xn, yn)) =∫P (yn+1|xn+1,
θ)P (θ|(x1, y1), (x2, y2), ..., (xn, yn))dθ.
(1.5)
Here, θ, stands for the network parameters (weights and biases)
and the likelihood (defined in
equation 1.1) is modified as:
L(θ|(x1, y1), (x2, y2), ..., (xn, yn)) =n∏i=1
P (yi|xi, θ). (1.6)
The component of yn+1 is estimated to be the mean of its
predictive distribution and defined as:
yn+1k =
∫fk(x
n+1, θ)P (θ|(x1, y1), (x2, y2), ...., (xn, yn))dθ. (1.7)
where fk, the network output functions, are relying on the
network parameters θ. A computationally
efficient Markov Chain Monte Carlo (MCMC) method is used to
compute the above integration.
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1.5 Markov Chain Monte Carlo integration (MCMC)
Bayesian prediction may take the form as in equation 1.5 or of a
single-valued prediction as in
equation 1.7. In both cases, the expectation value of the
function is computed relative to the
posterior probability density for the parameter (say A(θ)). The
expectation value of a continuous
random variable a(θ) is defined as:
E[a] =
∫a(θ)A(θ)dθ. (1.8)
Expectation values defined in equation 1.8 are approximated by
the Monte Carlo method, taking a
sample from A:
E[a] ≈ 1N
N∑t=1
a(θt), (1.9)
The Monte Carlo integration formula of equation 1.9 gives an
unbiased estimate of E(a) and
converges to an accurate value with increasing N.
1.6 Gibbs sampling
Any statistic can be computed from a posterior distribution, by
using equation 1.9, as long as
N simulated samples from that distribution is available. To do
this task, Gibbs sampling is one
MCMC technique, which is use to obtain a sample from the
posterior distribution. This technique
is applicable when sampling of one parameter is computed at a
time from a distribution conditioned
on all the other parameters. This is an iterative algorithm,
where the random sample of one variable
is drawn at a time from its conditional distribution with the
remaining variables fixed to their last
updated values. BANN hyperparameters are determined via Gibbs
sampling (as the conditional
distribution for a single hyperparameter is accessible). A
Hybrid MCMC algorithm is applied to
sample the weights and biases of the BANN.
1.7 Network architecture and prior computation
An ordinary network has zero or more hidden layers. The first
hidden layer is joined to the inputs,
and the rest of the layers are linked to the previous hidden
layer, and optionally to the input. The
output units are connected to the last hidden layer (and at
times to other hidden layers or input
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units). The prior distribution for the parameters of a network
is defined in terms of hyperparameters,
that control the standard deviation for weights and biases in
different groups. Neal [2012] defines
these hyperparameters in terms of “sigma values (σ)” and their
distribution are assigned in terms
of the respective “precision”, τ = σ−2, which are sampled from
Gamma distributions. Consider u1,
u2, ..., uk as the parameters in one group. The hyperparameter
for this group given the standard
deviations (σu) of a Gaussian prior is defined as:
P (u1, u2, ..., uk|σu) = 2(π)−k/2σukexp(−∑
ui2/2σu
2). (1.10)
τu is the precision and is defined as τu = σu−2.
The prior for the hyperparameter is expressed as (with mean
wu):
P (τu) =(αu/2wu)
αu/2
Γ(αu/2)(τu)
αu/2−1exp(−τuαu/2wu). (1.11)
The prior for τu is controlled by the values of αu (positive).
The process of computing the prior is
hierarchical with a three-level approach. At the base level,
each parameter is assigned a Gaussian
distribution with zero mean associated with some precision. At
the following level, the precision is
sampled from a Gamma distribution with a particular shape
parameter and with a mean given by a
hyperparameter (common to all parameters of the same subgroup).
The high-level hyperparameter
is chosen from a Gamma distribution with a specified mean and
with a specified shape parameter.
When training data is obtained, the prior is updated to a
posterior parameter distribution and is
then used to make predictions for a test case. A similar
three-level approach is used to fit the noise
levels.
1.8 Automatic Relevance Determination (ARD)
The number of predictors used in modelling the distribution of a
predictand is a source of complexity
in ANNs. Including more and more inputs leads to poor predictive
performance since irrelevant
inputs will, by chance, appear in the finite training set to be
more closely associated with the
targets than are the truly relevant inputs. So the number of
input variables must be limited, based
on estimation of which attributes are most likely to be
relevant. To accomplish this in BANNs,
individual hyper-parameters weight the contribution of inputs to
the network to maximize predictive
performance. Neal [2012] describes this process as “Automatic
Relevance Detection” (ARD). In
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ARD, each input variable has an associated hyperparameter that
controls the magnitude of the
weight on the connection out of that input unit. If the
hyperparameter that is associated with an
input identifies a small standard deviation (around zero) for
weight out of that input, these weights
will likely be small, and the input will have little effect on
the output, whereas, the significance of
the input will increase if the hyperparameter specifies a large
standard deviation.
1.9 Climate Turing Test
In the research of artificial intelligence, the Turing Test is
performed to determine if a machine’s
ability to exhibit intelligent behaviour is indistinguishable
from that of a human. The machine will
pass the test if it is intelligent and responsive in a manner
that is indistinguishable from a human. In
this project, the Turing test concept is implemented as a
Climate Turing test, to ascertain if the pre
glacial maximum behaviour (120 ka to 22.05 ka) of climate
predicted by our CG is indistinguishable
from the FAMOUS climate model. To execute the Climate Turing
test, the difference between CCSM
and FAMOUS is taken as a reference uncertainty to compare the
difference between the CG and
FAMOUS. The comparisons are made based on the mean correlation
(over space and time), mean
deviation, Root Mean Square Error (RMSE), map plots and
projections of Empirical Orthogonal
Function (EOF) of our CG predictions with FAMOUS climatological
simulated fields (monthly
mean temperature and precipitation). To pass the Climate Turing
test, simulated fields should have
relatively high correlation (i.e. with respect to GCM target
field), small RMSE, close patterns and
a reasonable capacity to capture natural variability compared to
the reference uncertainty.
1.10 Climate Generator (CG)
1.10.1 Reasoning behind the name of CG
Weather Generators (WGs) are computationally useful statistical
tools that can be used to generate
realistic and rapid daily sequences of atmospheric variables,
such as temperature and precipitation,
on a small scale. WGs are a means of generating a random time
series of ’weather’ that replicates
observed statistics. It can be used to investigate small-scale
climate impacts and to compute nu-
merous random realizations quickly. Moreover, WG outputs are set
to the observed distributional
20
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properties, primarily on the daily or sub-daily scale [Ailliot
et al., 2015]. In this research, we im-
plement the Weather Generator concept on a large
spatial-temporal scale and subsequently propose
the term Climate Generator (CG).
1.10.2 CG predictors
The climate of a geographic location has various strong
dependencies, such as latitude, earth-sun re-
lationships, proximity to large bodies of water, atmospheric and
oceanic circulation, topography and
local features. The climatological temperature field is
relatively smooth, depending most strongly on
latitude, surface elevation and continental position. However,
the climatological precipitation field is
not smooth and has strong longitudinal and non-local dependence.
Like a WG, our CG generates a
synthetic climatology conditioned on various inputs. In
probabilistic terminology, the CG provides a
posterior distribution for climate prediction conditioned on the
given predictors. Moreover, our CG
predicts temperature and precipitation fields jointly, so
predictands are correlated with each other.
The CG presented herein predicts monthly mean surface
temperature and precipitation fields by con-
sidering the above characteristic of climate through predictor
variables: latitude, longitude, monthly
mean sea surface temperature field from a fast low-resolution
Energy Balance Model (EBM), surface
elevation, ice mask, atmospheric concentrations of carbon
dioxide and methane, and orbital forcing.
Some predictor variables (latitude, longitude, Carbon dioxide
etc.) are already taken into account
by the EBM, but perhaps to an inadequate extent. We therefore
explicitly include those predictor
variables to create our CG and then test whether their inclusion
is required via ARD. Latitude and
longitude are included in our study because they have direct
effects on climate prediction. We select
surface elevation as a predictor due to strong vertical
temperature gradients that vary laterally. Due
to its high albedo, the presence of ice is also included as a
predictor. Carbon dioxide and methane
data are considered as a predictor given their relatively
significant radiative forcing variations over a
glacial cycle. Finally, we include monthly mean sea surface EBM
temperature in our predictor sets,
since the EBM can estimate the sea level monthly mean surface
temperature very efficiently. As an
attempt to capture non-local effects, the first two EOFs of
surface elevation and the area of ice were
tested in our CG as predictors but did not yield any significant
improvement in CG predictions.
21
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1.10.3 Climate Generator and reality
Computational tools, which are used in climate variability
representation, may have many sources
of uncertainty. For example, GCMs uncertainties are classified
into initial conditions, boundary con-
ditions, parametric and model structure. For the hypothetical
case of accurate initial and boundary
conditions, and accurate model parameter values, the model
structural uncertainty is then isolated:
Reality (t) = GCMs(t) + α, where α is the structural error. The
CG emulates GCMs, so all
GCMs uncertainties listed above propagate into the CG. The BANN
also estimates its own regress-
ive uncertainty. We make the assumption that this regressive
uncertainty is largely due to smaller
spatio-temporal scale dynamics and non-local couplings within
the GCM and thereby consider it as
climate noise. To capture GCM variance, this predictive
uncertainty of the BANN is used to specify
the variance of uncorrelated Gaussian noise that is added to
each CG prediction. This uncorrelated
aspect of the injected “climate noise” is a further source of
error. We did test the inclusion of the
first two surface elevation EOFs in the CG input set, but no
significant improvements arose. This
CG does not account for the structural error of the GCM (which
would be a PhD Thesis in itself).
1.10.4 Present setup and future prospect
The CG is tested for two months: February (the coldest month)
and August (the warmest month).
The CG is trained with FAMOUS climate model data (rename as
CGfamous) for the time period
22 ka to present. To independently test this approach, CG
predictions are compared against a 98
kyr interval of GCM (FAMOUS) outputs that was not used for CG
training. The CG (BANN) was
then retrained with the CCSM (a much better GCM than FAMOUS) on
same training interval and
rename as CGccsm. The CG is a computationally efficient
(simulation of 120 ka to 22.05 ka took
15 minutes, approximately the same computationally speed as the
EBM) stochastic climate model.
The CG will be coupled with a 3D ice sheet model for glacial
cycle modelling. The CG approach
could also be applied to more advanced EMICs (e.g. LOVECLIM) for
climate generators on short
time scales.
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1.11 Summary
The objective of this research is therefore:
To create a computationally efficient stochastic climate model
(involving in part spatio-temporal
downscaling) to simulate atmospheric fields over last glacial
cycle timescales that are indistinguish-
able (for a specified set of metrics) to the output of GCMs.
1.12 Thesis Overview
This thesis is written in MUN manuscript format. This chapter
provides a review of the literature
and background materials for the article that is presented in
chapter two. More CG (BANN) outouts
comparison with FAMOUS, CCSM and EBM climate model outputs are
added in the Appendix.
23
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Chapter 2
Stochastic climate representation
for millennial scale integration over
North America
2.1 Abstract
This paper presents a computationally efficient stochastic
climate model to simulate atmospheric
fields (specifically monthly mean temperature and precipitation)
on large spatial-temporal scales.
In analogy with Weather Generators (WG), the model can be
considered a “Climate Generator”
(CG). The CG can also be understood as a field-specific General
Circulation climate Model (GCM)
emulator. It as invokes aspects of spatio-temporal downscaling,
in this case of a T21 Energy Balance
climate Model (EBM) to a GCM scale. The CG produces a synthetic
climatology conditioned on
various inputs. These inputs include sea level temperature from
a fast low-resolution EBM, surface
elevation, ice mask, atmospheric concentrations of carbon
dioxide, methane, orbital forcing, latitude
and longitude. Bayesian Artificial Neural Networks (BANN) are
used for nonlinear regression against
GCM output over North America. Herein we detail and validate the
methodology. To impose
natural variability in the CG (to make the CG indistinguishable
from a GCM) stochastic noise is
added to each prediction. This noise is generated from a normal
distribution with standard deviation
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computed from the 10% and 90% quantiles of the predictive
distribution values from the BANNs for
each time step. This derives from a key working
assumption/approximation that the uncertainty in
our prediction is in good part due to the “noise” of the GCM
climate. Our CG is trained against
GCM (FAMOUS and CCSM) output for the last deglacial interval (22
ka to present year). For
predictive testing, we compare the CG predictions against GCM
(FAMOUS) output for the disjoint
remainder last glacial interval (120 ka to 22.05 ka). The first
application of the CG will be for glacial
cycle modelling with the Glacial Systems Model.
2.2 keywords
Climate Generator, Weather Generator, Climatology, BANNs
2.3 Introduction
A Weather Generator (WG) generates a synthetic time series of
weather data for a location based
on the statistical characteristics of the observed weather. A WG
generally operates on a small scale,
is computationally efficient to enable numerous random
realizations, and its output is designed to
have the same distributional properties as the observed time
series, usually on the daily or sub-
daily scale [Ailliot et al., 2015]. In contrast, climate models
reproduce the behaviour of the whole
atmosphere and its interactions with the other components of the
Earth system (oceans, vegetation,
etc.) on a large spatial scale and generally for longer-term
intervals. Here, we expand the WG concept
to a large spatial-temporal scale, proposing the term Climate
Generator (CG) for the expansion.
Our primary aim is to create a fast, efficient stochastic
climate representation for glacial cycle scale
modelling. General Circulation Models (GCMs) are presently too
computationally expensive for such
contexts. For example, the widely used Community Climate System
Model (CCSM3) took about 2
years on 100 cores to simulate the 22 ka to present [Liu et al.,
2009]. Earth System Models of Inter-
mediate Complexity (EMICSs) are more appropriate for ten kyr
scale climate simulations [Claussen
et al., 2002], but with varying tradeoffs between resolution,
accuracy of climate representation, and
computational speed. Energy Balance climate Models (EBMs) are
the fastest EMICS for a given
resolution but, due to their lack of atmospheric dynamics and
precipitation fields, are inadequate for
glacial cycle modelling [Hyde et al., 1990], [Tarasov and
Peltier, 1997]. To date, most glacial cycle
25
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ice sheet simulations use a glacial index (for example Tarasov
et al. [2012]) in a combination with
LGM timeslice GCM output for their climate forcing.
Unless full energy-balanced surface mass-balance is being
computed, ice-sheet models for glacial cycle
contexts do not require accurate year to year prediction in
their climate forcing. The dynamical
memory of ice sheets filters interannual fluctuations. As such,
30 to 50 year scale monthly mean
climatologies with statistics for shorter scale temperature
variability are a reasonable trade-off for
enabling computation over such timescales. Given this context,
we invoke the concept of a Climate
Generator (CG) for improving the output of fast EMICS (in this
case a geographically resolving 2D
EBM) for 100 kyr scale contexts. The CG is a large
spatio-temporal scale dynamic representation of
climate based on a regressed relationship between the EBM (and
various other predictor variables)
and relevant GCM outputs.
To build up such a relationship, Artificial Neural Networks
(ANNs) are common in small spatial-
temporal scale climate prediction [Schoof and Pryor, 2001]. ANNs
have the potential for complex
non-linear input-output mapping [Dibike and Coulibaly, 2006].
However, ANNs do not have asso-
ciated uncertainty estimates, and over-fitting is a hazard. To
minimize over-fitting and to find an
optimum network, ANNs rely on a cross-validation test.
Cross-validation does not use training data
efficiently as it requires disjoint data sets for testing and
parameter estimation. Bayesian Artificial
Neural Networks (BANNs) generate uncertainty estimates and avoid
the need for cross-validation.
In BANNs, an assumed prior distribution of parameters (weight
and biases) is used to specify the
probabilistic relationship between inputs and outputs. The prior
distribution is updated to a pos-
terior distribution by a likelihood function through Bayes
theorem. The predictive distribution of
the network output is acquired by integration over the posterior
distribution of weights. BANNs
are used in different applications e.g., to create weather
generators [Hauser and Demirov, 2013],
for model calibration [Hauser et al., 2012], and for short time
scale climate prediction [Maiti et al.,
2013], [Luo et al., 2013]. Our CG uses BANNs to estimate a
posterior distribution for climate
prediction/retrodiction conditioned on various inputs including
the output of an EBM.
2.4 Test study region and data
The test study region of interest is North America (including
Greenland; more specifically, longitude
188E : 355E and latitude 34N : 86N). This combines a continent
that experience past ice cover but
no significant present-day ice cover with a region that has had
continued ice cover to present. We
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train our CG (CGccsm) against the full-time interval of the
CCSM3 (Monthly means, T31) climate
model [Liu et al., 2009] radiative surface temperature and
precipitation monthly mean output. For
methodological validation, we use the FAMOUS (low-resolution
GCM, atmospheric part of the
model has resolution 5 ◦ × 7.5 ◦) climate model data sets [Smith
and Gregory, 2012] (1.5 meter air
temperature and precipitation) and train our CG (CGfamous)
against the same time interval as
CCSM. The FAMOUS model was run with an accelerated mode (factor
10) for the full Last Glacial
Cycle (LGC) (120 ka to present). We divide the full-time
interval into two parts: the training
interval is 22 ka to the present year and the test interval is
120 ka to 22.05 ka. For this initial test of
concept, the CG is implemented for the coldest month (February)
and the warmest month (August),
with 50 year climatologies (5 consecutive years for FAMOUS and 5
years spaced 10 years apart for
CCSM) for model calibration and validation. Each month has 2400
time steps for FAMOUS (120
ka to present year) and 440 time steps for CCSM3 (22 ka to
present year). Each time step has 231
(21x11) gridcells (FAMOUS) and 602 (43x14) gridcell (CCSM) data
(for each predictors) and our
CG is trained to predict on each gridcell. For model validation,
we compare the CGfamous output
directly to that of FAMOUS over the test interval. These data
sets are not used for the CGfamous
training. For further evaluation, we compare our CGfamous,
CGccsm outputs with FAMOUS, EBM
(T11 @1500 km) and CCSM3 predictions over the last glacial
(test) period. For the comparisons,
EBM output [Deblonde et al., 1992] sea level temperature is
adjusted to surface temperature with
a lapse rate of 6.5 K/km, as this has been standard practice for
using EBM climate models. EBMs
generally do not take into account the spatio-temporal variation
of vertical temperature gradients.
2.5 Methods
The CG uses Bayesian Artificial Neural Networks (BANNs) for
estimating an evolving climate state
as a function of various inputs. The BANNs also estimate
predictive uncertainty which (in an ar-
guable leap) we take to represent the shorter scale un-resolved
variability in the climate. BANNs
are effectively a set of artificial neural networks with
individual parameters from a posterior prob-
ability distribution derived from training the network against
observed input-output sets. The CG
estimates target values based on the mean from the resulting set
of networks, and its squared error.
Further parameter sampling results from creating several network
sets with distinct initial seeds.
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2.5.1 BANN design and training
We design our ANN architecture by using the software for
flexible Bayesian modelling package
(freely available) at
http://www.cs.utoronto.ca/radford/fbm.software.html. Different
architectures
(number of hidden layers, node size, connection of inputs,
hidden units and outputs) and differ-
ent predictors are first tested. Architectures are selected
according to the predictive skill on the
test data. Step size and prior specification are then adjusted
to improve prediction capability. To
choose an appropriate predictor set, we tested the following
predictor variables: latitude, longitude,
EBM sea level temperature (Deblonde et al. [1992]), carbon
dioxide (Luthi et al. [2008]), meth-
ane (Loulergue et al. [2008]), surface elevation [Smith and
Gregory, 2012], surface type (ice) [Smith
and Gregory, 2012], orbital forcing (came from EBM) in different
time and location, temperature
and precipitation ([Smith and Gregory, 2012] and [Liu et al.,
2009]), melt water flux (five different
locations; [Smith and Gregory, 2012]), first two EOFs of ice
surface elevation, and ice area. The
outputs are monthly mean surface temperature and precipitation.
Various network architectures
and different combinations of predictor variables were tested.
Automatic Relevance Determination
[ARD] Neal [2012] was used to identify which predictors provide
meaningful weight in the distri-
bution value. Various combinations of the input set and network
architectures were also evaluated
against the test interval subset of the GCM output (as detailed
below). The resultant optimal input
set is comprised of: latitude, longitude, surface elevation,
ice, carbon dioxide, methane, orbital for-
cing (June/July/August mean solar insolation at 60N), and EBM
sea level temperature. The best
fitting BANN architecture has two hidden layers of tanh(x)
hidden unit as detailed in Figure 2.1.
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Figure 2.1: Final ANN Diagram (Architecture type I) used in
CGfamous and CGccsm
prediction
2.5.2 BANN Implementation
Bayesian Artificial Neural Networks (BANNs) estimate a
probability distribution and are derived
by training the network against available data using Bayesian
inference. Markov Chain Monte Carlo
(MCMC) sampling is used for selecting the distribution
parameters for the networks. The step by
step procedure for BANN’s implementation follows Neal
[2012]:
• Process predictor and predictand data sets.
• Define architecture and specify model to have real valued
targets.
• Define a prior distribution for parameters (weight and
biases). The prior is specified hier-
archically for a group of parameters with a three-level approach
as follows. Each prior is
sampled from a Gaussian distribution with zero mean and some
precision (inverse standard
deviation). The precision for a “sub-group” of parameteris in
turn selected from a Gamma
distribution with a selected shape parameter and with a mean
given by a hyperparameter.
Finally, this hyperparameter is sampled from a Gamma
distribution with a specified mean
and with another assigned shape parameter. Mean precision at the
top level is assigned as 0.1
(width) and the shape parameters of the Gamma distribution are
assigned a value of 2. Priors
for input to hidden layer, connections between hidden layers and
hidden layers to outputs are
automatically re-scaled based on the number of hidden units and
the prior of the output bias
is specified as a Gaussian prior with mean zero and standard
deviation 10.
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• A noise model is fitted to estimate network parameters. To fit
a noise model each prediction
for targets is considered to be a sample from a Gaussian
distribution. The noise levels
specification follows the same three level approach used for
specification of the priors. The
targets are modelled as the network outputs plus Gaussian
noise.
• Specify data for training and testing.
• Initialize the network with hyperparameters set to (say) 0.3
and all parameters set to zero.
Markov chain operations are defined, where each iteration
consists of order fifty repetitions
of the following steps: Gibbs sampling for the noise level,
hybrid Monte Carlo sampling using
a fixed trajectory and step size adjustment factor. In this
stage, the hyperparameters are not
updated and remain fixed at a value 0.3.
• A single iteration of the above process is representative of
one step in Markov chain simulation.
The rejection rate is examined after a number of (say 50) hybrid
Monte Carlo updates. If the
rejection rate is high (say, over 40%), the Markov chain
simulation is repeated with a smaller
step size adjustment factor.
• a network is stored in the log file containing the parameters
and hyperparameters values.
Markov chain sampling is repeated and overrides the previous
set. Each iteration consists of
say five repetitions of the following steps: Gibbs sampling for
both the hyperparameters and
the noise level followed by hybrid Monte Carlo sampling as
above. (A long trajectory length
is useful for the sampling phase).
• By looking at the hyperparameters values and quantities such
as the squared error on the
training set, we can get an idea of when the simulation has
reached equilibrium or not. After
that, we can start prediction.
• Generate predictions (mean, 10% and 90% quantiles) for the
test cases from the resultant
distribution of networks. By using different initial seeds,
ensembles of several networks are
generated by sub-sampling from the later segments of the Markov
Chains.
The detailed implementation procedure is given in Neal
[2012].
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2.5.3 Adding noise
Gaussian noise is added in our CG prediction to account for (at
least in part) seasonal to decadal
climatic variability not captured by the EBM. This natural
variability (noise) is physically correlated
across space and time. But given the context (coupling with ice
sheet models for glacial cycle times-
cales) and for computational simplicity, the CG noise injection
uses uncorrelated random sampling.
Ice sheet thermodynamic response to climate is smoothed to
centennial or longer timescales. Surface
mass-balance response for the given grid scales will be
sensitive to the variance of temperature but
not to spatial correlations nor much to temporal correlations.
Correlations between temperature
and precipitation could have significant impact, especially
during the potential melt season. The
August residual (CG without noise - FAMOUS) correlation map
between the temperature and pre-
cipitation fields shows magnitudes of mostly less 0.3 (Fig. A.18
in the Appendix) and are therefore
relatively small (this does not, however, rule out significant
non-linear relationships). The random
noise is added to each time step of our CG predictions by
generating a random sample from Gaussian
distribution with ∼ N(µ = 0, σ), where µ = Mean and σ = Standard
deviation (80% confidence
interval scale). The standard deviation is computed through the
BANNs predicted 10th percentile
and 90th percentile of the predictive distribution of a single
guess for each case. Standard deviation
is computed from the following Equation:
σ =(X90% − (X90% +X10%)/2)
Z90%(2.1)
where Z90% = 1.28 (Z values or score), calculated from the
statistical table. The values of σ,
defined in the Equation 2.1 had space and time dependence. This
assumption that BANN predictive
uncertainty can provide an approximate estimate for the
unresolved climatic variability is tested in
part below.
2.6 Implementation Results
This research implements the “Turing Test” concept as a “Climate
Turing Test (CTT)” to measure
the prediction capability of our CG. The CTT determines whether
or not CGfamous is capable of
predictions like the FAMOUS climate model. To implement the CTT
concept, a direct comparison
was done between CGfamous and FAMOUS Climate model outputs over
the test interval (120 ka
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to 22.05 ka) using characteristics such as Root Mean Square
Error (RMSE), visual patterns in map
plots and Empirical Orthogonal Function (EOF) projection to
measure the difference in outputs
between CGfamous and FAMOUS relative to that between CCSM and
FAMOUS. Here we give
some example comparisons for different architectures and
predictor sets.
2.6.1 Selection of BANN architecture
More than 100 different BANN architectures (different number of
hidden layers, different connections
and node sizes) with different combinations of predictor sets
were tested. To convey the sensitivity
to architecture and predictor set, we present results for three
basic architectures (Figure 2.1 and
Figure 2.2) in combination with various predictor sets (Table
2.1). Architecture type I (Figure 2.1)
gave the best overall fit of CGfamous to FAMOUS over the test
interval.
(a) Architecture type II (b) Architecture type III
Figure 2.2: ANN Diagrams are tested to improve CG prediction
Table 2.1: Predictor sets for displayed CG results
Combination 1: PS1 = Latitude, Longitude, surface elevation,
ice, EBM temp
Combination 2: PS2 = PS1, CO2 + CH4, orbital forcing, melt water
flux
Combination 3: PS3 = PS1, carbon dioxide, methane, orbital
forcing
Combination 4: PS4 = PS3, EOF-1 and EOF-2 of ice data, ice
volume
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Table 2.2: Different models based on architecture and inputs
Model Name Predictors Architecture type
A PS3 II
B PS1 I
C PS3 III
D PS2 I
E PS4 I
F PS3 I
Six model names are assigned in Table 2.2 based on different
predictor sets and architecture (Table 2.1).
Table 2.3: MEAN DIFFERENCE (MD) and RMSE relative to the FAMOUS
(except for
last entry) over the full grid area (train period)
TEMP (DEG C) PREC (cm/month)
FEB AUG FEB AUG
Model MD RMSE MD RMSE MD RMSE MD RMSE
A 3.91 9.07 4.09 5.7 0 3 1 3
B 2.85 10.19 -2.93 9.9 1 4 2 5
C 2.43 7.94 5.45 5.36 1 3 1 4
D -3.3 4.13 -1.73 2.67 0 2 0 2
E -2.35 5.77 -1.05 4.6 0 2 1 3
F -0.45 3.92 -1.23 2.52 0.5 3 0 3.1
M (1) 5.87 10.51 -4.4 4.29 0.02 3.08 0 3.06
CCSM (2) 5.86 10.99 -4.4 4.08 0.5 4.1 -1.3 4.1
EBM 12.46 11.0 -1.52 4.77
(1) versus (2) -0.02 4.14 0.01 2.67 0.5 4.1 -1.3 4.1
33
-
These models are compared with FAMOUS model outputs over the
training interval in Table 2.3
and over the test interval in Table 2.4. The positions of each
letter appearing in Figure 2.3 and
2.4 quantifies how closely that model’s simulated temperature
and precipitation pattern match the
FAMOUS climate model outputs and gives a graphical summary of
comparisons of the RMSE and
Standard deviation. RMSE is computed from the differences
between different CG predictions
with FAMOUS climate model outputs. The Standard deviation of
FAMOUS is indicated by the
black contour line in Taylor diagrams. Our main selection
criterion is the minimization of RMSE.
Beside this, mean deviation (general bias) comparison on Table
(2.4) and standard deviation and
correlation from Figure 2.3 and Figure 2.4 allow us to see how
comparable our models prediction are
with FAMOUS. The central RMSE is about 2.98 ◦C (August
temperature) and 5.49 ◦C (February
temperature) for model F , which are the lowest compared to all
other models in Table 2.4. In the
case of precipitation, the RMSE of model F is about 1.9 cm/month
(February) and 2.8 cm/month
(August), which also are the least compared to all other
models.
Table 2.4: MEAN DIFFERENCE (MD) and RMSE relative to the FAMOUS
(Test period)
over the full grid area
TEMP (DEG C) PREC (cm/month)
FEB AUG FEB AUG
Model MD RMSE MD RMSE MD RMSE MD RMSE
A 4.45 8.08 3.93 9.1 -0.16 2.5 1.07 4
B 3.35 9.11 -3.09 8.5 0.83 3 2.07 5
C 3.2 6.49 5.29 12.5 -0.18 5 0.07 3
D -2.75 5.74 -1.29 5.22 -0.16 2.3 -0.93 3
E -1.75 5.38 -1.2 4.68 -0.15 2.3 0.07 3.4
F -2.45 5.49 -1.45 2.98 -0.15 1.9 0.07 2.8
M -4.26 11.27 -6.64 5.36 0 3 0.03 3.3
EBM 10.82 10.45 -5.3 4.76
34
-
February Temperature (Deg C)
Standard deviation (Deg C)
Sta
ndard
devia
tion (
Deg C
)
0 10 20 30 40
010
20
30
40
10
20
30
40
0.10.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.99
Correlation
A
B
C
DE F
M
FAMOUS
February Precipitation (CM / Month)
Standard deviation (CM / Month)
Sta
ndard
devia
tion (
CM
/ M
onth
)
0 2 4 6
02
46
2
4
6
0.10.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.99
Correlation
A
B
C
D
E
F
M
FAMOUS
Figure 2.3: Taylor diagram displaying a statistical comparison
with FAMOUS (test part) of
six different CG estimates. Pattern similarities are quantified
based on their correlation and
centered root mean square difference between CG and FAMOUS, and
standard deviation
with respect to the mean of the corresponding field. Contour
grey lines indicate the root
mean square (RMS) values. The model M is the same as model F but
trained with CCSM
climate model data.
35
-
August Temperature (Deg C)
Standard deviation (Deg C)
Sta
ndard
devia
tion (
Deg C
)
0 5 10 15 20
05
10
15
20
5
10
15
20
0.10.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.99
Correlation
A
B
C
D
E
F
M
FAMOUS
August Precipitation (CM / Month)
Standard deviation (CM / Month)
Sta
ndard
devia
tion (
CM
/ M
onth
)
0 1 2 3 4 5
01
23
45
2
4
0.10.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.95
0.99
Correlation
A
B
C
D
E
F
M
FAMOUS
Figure 2.4: Taylor diagram displaying a statistical comparison
with FAMOUS (test part) of
six different CG estimates. Pattern similarities are quantified
based on their correlation and
centered root mean square difference between CG and FAMOUS, and
standard deviation
with respect to the mean of the corresponding field. Contours
grey line indicates the root
mean square (RMS) values. The model M is the same as model F but
trained with CCSM
climate model data.
36
-
The model F has strong mean correlation (over space and time)
with FAMOUS at about 0.98 for
February temperature and 0.92 for February precipitation in
Figure (2.3 and 2.4). The model F has
the best fit compared to other models listed in Table 2.4. As is
evident in Tables 2.3 and 2.4 and
figures (2.3 and 2.4), there is significant sensitivity to
network architecture and predictor set with
a factor 4 range in temperature RMSE and factor 2 range in
precipitation RMSE for the example
combinations. The sensitivity to predictor sets is quantified in
more detail through ARD as described
below.To choose predictor sets, an ARD analysis has been done.
The ARD test helped us to choose
which inputs are relevant for the outputs and is determined
based on the hyperparameters values
which control the standard deviation for weights and biases in
different groups. In Neal [2012],
these hyperparameters values are referred to as “Sigma” values.
The significance of each input is
non-linear proportional to the Sigma value in Table 2.5.
Table 2.5: ARD analysis for the Model F (CGfamous)
Predictors Sigma value
Latitude 1.30
Longitude 0.67
Elevation 2.85
Ice 2.79
CO2+CH4 1.18
Orbital forcing 3.42
EBM temp 4.46
For the case of Model F , EBM temperature is the most
significant input. However, even though
the EBM computes orbital forcing and accounts for greenhouse
gases, orbital forcing is still the next
most significant input. Longitude is the least important, even
though temperature and precipitation
have in reality high dependence on longitude given atmospheric
circulation dependencies. The EBM
does have a slab ocean with thermodynamic sea ice, and this
result suggests that continentality
effects might be reasonably captured by the Model EBM. Further
comparisons were carried out
with field plots and EOF projections for all simulated fields.
Model F predictions have the least
37
-
RMSE (compared to FAMOUS) among all tested models. It has the
highest positive correlation and
comparable mean difference (with FAMOUS). Therefore the model F
has been chosen and renamed
CGfamous.
2.6.2 Model Comparison
Weather Generator performance can be tested against new
observations, but such opportunities
are limited in our case. The CG is evaluated based on computing
statistics (RMSE, Mean devi-
ation, correlation), the goodness of fit (including noise
levels) and qualitative consideration against
FAMOUS climate model outputs (test interval). The best and worse
fits are identified between CG
prediction and FAMOUS outputs on specific regions or latitude
bands, winter versus summer and
full grid area versus ice region. The comparison over the ice
region gives the opportunity to check the
prediction capability of our CG in the context of glacial cycle
modelling. Criteria such as space-time
scale appropriateness, patterns, and climate noise variability
at shorter time scales are introduced to
measure variance. The difference between CCSM and FAMOUS over
the training interval is taken
as a minimum value of model uncertainty and thereby our
reference misfit bound for the climate
Turing Test.
Over both the test and training interval, the RMSE of CGfamous
simulated temperature field (about
FAMOUS) is about 50% less than the RMSE of CCSM and EBM model
(about FAMOUS) over the
full grid area. For precipitation over the test interval, the
RMSE of CGfamous (about FAMOUS)
is approximately 42% less than the RMSE of CCSM (CCSM versus
FAMOUS). The CG simulated
August temperature has a better fit with FAMOUS (less RMSE)
compared to that of February.
Also, CG seems to work better in the ice region (approximately
54% or more less RMSE compare to
the full grid area). Besides these quantitative statistics,
CGfamous also has improved the geographic
pattern of misfits (CGfamous -FAMOUS versus FAMOUS - CCSM and
FAMOUS - EBM). The CG-
famous simulated temperature field (both months) has a clear
cold bias (in test part) compared to
FAMOUS. But CG CCSM has a sharp and warm bias (February) in
addition to a clear cold bias
(August) compared to the FAMOUS test map. The CG simulated
temperature field has captured
approximately 70% temporal variance (based on the leading two
EOFs) of that of FAMOUS. Pre-
cipitatation is a challenge for all models, and the CGfamous
precipitation field captures about only
40% (based on leading two EOFs) of the temporal variance
compared to FAMOUS. For context, the
CCSM precipitation field is also far from that of FAMOUS.
38
-
For both months over the training period, the CGfamous (network
F ) temperature RMSE relative
to FAMOUS is less than half of our structural uncertainty
reference (i.e. CCSM - FAMOUS) and
the corresponding precipitation RMSE is about 33% smaller (TABLE
2.4). The temperature RSME
not unexpectedly increases over the predictive test region, but
critically these values are still about
half of the CCSM - FAMOUS reference RMSE. CGfamous temperature
is highly correlated with
FAMOUS (0.94 or higher) over the test interval with a
temperature bias (MD) that is 2 degree
(about 40%) lower than the CCSM bias over the test interval
(Table 2.4). As structural uncertainty
is larger than the difference between CCSM and FAMOUS, CGfamous
temperature largely passes
the Climate Turing Test for these metrics. The approximate
factor 3 improvement in RMSE of
CGfamous versus that of the EBM validates the CG role of
statistical improvement of the fast
simplified models.
Table 2.6: Mean Difference (MD), Correlation (COR) and RMSE
relative to FAMOUS
(except for last entry) over the ice region (Train period)
TEMP (Deg C) PREC (cm/month)
FEB AUG FEB AUG
Model MD RMSE COR MD RMSE COR MD RMSE COR MD RMSE COR
EBM 1.58 4.41 .99 -0.38 1.76 .98
CGfamous 0.11 1.33 1 0.02 0.53 1 0.05 0.41 .87 0.01 0.92 .95
CGccsm(1) 1.06 3.40 .99 -1.23 1.24 .99 0.06 0.38 .86 0.01 0.89
.96
CCSM(2) 1.06 3.58 .99 -0.27 1.19 .99 -0.01 0.39 .87 -0.16 0.78
.96
(1) versus (2) 0 0.92 .99 -0.02 0.51 1 0.02 0.30 .86 0.16 0.54
.96
Table 2.7: Mean Difference (MD) , Correlation (COR) and RMSE
relative to FAMOUS
(Test period) over the ice region
TEMP (Deg C) PREC (cm/month)
FEB AUG FEB AUG
Model MD RMSE COR MD RMSE COR MD RMSE COR MD RMSE COR
EBM 1.68 4.24 .99 -0.83 2.21 .93
CGfamous -0.37 1.79 .99 -0.26 1.10 .96 0.02 0.40 .75 -0.20 0.94
.89
CGccsm 1.02 3.2 .98 -0.68 1.94 .94 0.02 0.31 .77 -0.08 0.78
.88
39
-
For precipitation over the test interval, the RMSE values of
CGfamous are about 53% smaller
(February: 1.9 versus 0.41 cm/month) and 31% smaller (August:
2.8 versus 4.1 cm/month) than the
RMSE of CCSM precipitation (Table 2.4). Except for the weaker
correlation for August compared
to February (0.72 versus 0.92 cm/month) over the test interval
(arguably again within structural
uncertainty), CGfamous precipitation also passes this components
of the climate Turing test.
(a) February (Full grid area) (b) February (Ice region)
(c) August (Full grid area) (d) August (Ice region)
Figure 2.5: Comparison of the spatial mean (with latitudinal
weighting) temperature time
series. The black vertical line separates the test (left) and
training part (right).
40
-
Even with the increased complexity and higher resolution of
CCSM, there is little deterioration in
the training fit of CGccsm to CCSM compared to that of CGfamous
to FAMOUS. All statistics
(RMSE, MD and correlation) for CGfamous and CGccsm are better
when computed just over ice
covered regions (Table 2.6 and Table 2.7).
Figure 2.6: The August temperature field (Deg C) at 100 ka (1st
and 2nd row) with elevation
and ice contours shown in black and blue. The difference between
plots are shown in the
3rd row. Model names are indicated in the top left corner in
each box.
Our climate Turing test also requires assessment of map plot
timeslices. A 100 ka August temper-
ature fields comparison (Fig. 2.6) indicates regional biases,
with the most evident being a strong
41
-
cold bias over Greenland. However, the discrepancies are
significantly less than the 18 ka difference
between CCSM and FAMOUS (Fig. A.4 in the appendix). Furthermore,
there is no obvious visual
pattern that one could apriori use to ascertain which field was
from FAMOUS versus CGfamous.
Figure 2.7: February precipitation field (cm/month). Left
column: test case. Right column:
Training Case. Model names and times are indicated in the top
left corner in each box.
The geographic pattern of precipitation misfit between CGfamous
and FAMOUS at 18 ka is very
close to that of CCSM and FAMOUS (Fig. A.5). The misfit patterns
for example instances from
predictive and training regimes (100 ka and 20 ka in Fig. 2.7)
do not show obvious poorer predictive
42
-
capability of the CG for the 100 ka timeslice than for the 20 ka
training timeslice.
−120000 −100000 −80000 −60000 −40000 −20000 0
−4
02
46
EOF−1 55.33%
year
expn
−coe
f
−4 −2 0 2 4 6
0.0
0.1
0.2
0.3
0.4
0.5
EOF− 1 55.33 %
expn−coef
Den
sity
−120000 −100000 −80000 −60000 −40000 −20000 0
−6
−2
02
4
EOF−2 14.51%
year
exp
n−
co
ef
−6 −4 −2 0 2 4
0.0
0.1
0.2
0.3
0.4
0.5
EOF− 2 14.51 %
expn−coef
De
nsity
FAMOUS CGfamous EBM CCSM
Figure 2.8: Temperature field, left column: Distribution of the
expansion coefficient over
total time steps for the leading two EOFs of the FAMOUS Model
data (black), and the
distribution of the expansion coefficients over time obtained by
an ensemble projection of
CGfamous simulation (red), EBM temperature (blue) and CCSM
(cyan) onto the same
EOFs. Top time series represent the August EOFs and bottom time
series are the Feb-
ruary EOFs. Right column: Distribution of the expansion
coefficients for the leading two
FAMOUS EOFs for the FAMOUS (black), CGfamous (red), EBM
temperature (blue), and
CCSM (cyan) datasets
43
-
−120000 −100000 −80000 −60000 −40000 −20000 0
−4
−2
02
4
EOF−1 37.07%
year
expn−
coef
−4 −2 0 2 4
0.0
0.2
0.4
0.6
0.8
1.0
EOF−1 37.07%
expn−coef
Density
−120000 −100000 −80000 −60000 −40000 −20000 0
−6
−2
02
46
EOF−2 5.12%
year
exp
n−
co
ef
−6 −4 −2 0 2 4 6
0.0
0.2
0.4
0.6
0.8
1.0
EOF−2 5.12%
expn−coef
De
nsity
FAMOUS CGfamous CCSM
Figure 2.9: Precipitation field, left column: Distribution of
the expansion coefficient over
total time steps for the leading two EOFs of the FAMOUS Model
data (black), and the
distribution of the expansion coefficients over time obtained by
an ensemble projection of
CGfamous simulation (red) and CCSM (cyan) onto the same EOFs.
Top time series rep-
resent the August EOFs and bottom time series are the February
EOFs. Right column:
Distribution of the expansion coefficients for the leading two
FAMOUS EOFs for the FAM-
OUS (black), CGfamous (red) and CCSM (cyan) datasets
For spatial comparison over time, the CCSM simulated fields and
an ensemble of CGfamous sim-
ulations are projected onto the two leading EOFs of the FAMOUS
climate model (Figure 2.8 for
temperature and Figure 2.9 for precipitation). CCSM simulated
fields have less variance compared to
FAMOUS and the CGfamous simulated temperature field has better
fit with FAMOUS compared to
CCSM. The time evolution of the expansion coefficients in the
CGfamous simulated fields compared
44
-
to FAMOUS are comparatively better than that of CCSM. The 1st
EOF represents about 56% of
the simulated CGfamous temperature field, while the 2nd EOF
represents about 15% (Figure 2.8).
For the case of precipitation, only the first EOF is significant
(37%), again with a closer match to
FAMOUS than that of CCSM.
In summary, the CGfamous simulated temperature field better fits
FAMOUS compared to CCSM
versus FAMOUS and EBM versus FAMOUS and efficiently generates
ensembles which represent
large-scale climate variability. Even though the EBM has no
precipitaition field, CGfamous is still
able to capture FAMOUS precipitation fited to our structural
error (CCSM-FAMOUS) reference.
The above comparisons have led our CG to reasonably pass the
Climate Turing Test.
2.7 Discussion
We have constructed a computationally efficient climate model
(our Climate Generator or GCMs
emulator) as an alternative to expensive GCMs for 100 kyr scale
integrations. To compensate for
inadequate variance in the BANN output within the CG, Gaussian
noise is injected into our CG at
each time step with zero mean and spatially varying standard
deviation calculated from ensemble
networks prediction values (10% and 90% percentiles). The CG
takes about 15 minutes to generate
February and August climatologies over the 120 ka to 22 ka
interval at 50 year time steps. To
compare the prediction capability of our CG with the GCMs, we
introduced the Turing test concept
as a Climate Turing Test (CTT). To implement the CTT concept, a
direct comparison was done
between CGfamous and FAMOUS Climate model outputs (120 ka to 22
ka). These data sets are not
used in CG training. The difference between the CCSM and the
FAMOUS over the training interval
(20 ka to present year) are considered as a minimum structural
uncertainty estimate for both GCMs.
We take this uncertainty as a reference to determine whether the
CG passes the Climate Turing
Test over the 120 ka to 20 ka test interval.
The CGfamous simulated fields have a smaller RMSE relative to
FAMOUS (Temperature: about 50%
less and precipitation about 33% less) in the test part (Table
2.4) compared to the CCSM uncertainty
(RMSE) listed on (Table 2.3). CGfamous also has relatively
better fits over the ice region. CGfamous
extracts varying vertical temperature gradients given the
significantly reduced misfits over high
elevation regions compared to that of the EBM (Figure 2.6). The
main comparative deficiciency
is the significant dry bias over the Great Lakes (20 ka) and
east thereof at 100 ka (Figure 2.7).
45
-
The CGfamous simulated temperature field has a cold bias (over
the test interval) compared to
FAMOUS (February and August) while CGccsm has a cold bias
(August) and sharp warm bias
(February) compared with FAMOUS over the test interval. A
similar pattern of bias occurs over
the training interval (Figure 2.5). The FAMOUS output has more
variance compared to that of
CCSM (Figure 2.5) in part due to the lack of available matched
fields (only the radiative surface
temperature from the CCSM and the 1.5 meter air temperature from
FAMOUS were available).
The imperfection between climate generator predictions and
reality can be conceptually broken down
into two components. The first is the stochastic process error
between the CG and GCM and the
second is the structural error of the GCM relative to reality.
Our simulated precipitation field has
less variance compared to FAMOUS, and future development of the
CG will explore other predictor
sets which have relevance to precipitation prediction such as
hydrology components.
2.8 Conclusions
We have introduced the concept of a Climate Generator to create
a large spatio-temporal scale
climate representation for coupled ice sheet modelling over
glacial cycles. The CG expands the scale
of weather generators. For this proof of concept, the CG was
implemented over North America.
For validation, we compared CGfamous simulated fields against
FAMOUS simulated fields (over the
test interval which was not used for training the Bayesian
artificial neural networks in the CG). We
introduced the Climate Turing Test concept to provide a
pass/fail reference for field comparison.
The FAMOUS GCM was used for CG proof of concept/validation and
then the CG was retrained
against the much more advanced CCSM (CGccsm). CGfamous and
CGcssm have test and training
interval errors with respect to their corresponding GCMs that
are of the same scale (and mostly
less than) our minimal structural error estimate. This estimate
is based on the difference between
FAMOUS and CCSM temperature and precipitation fields. As such,
the CG passes the Climate
Turing test. It was not all a priori clear whether this would be
possible given the CG reliance on the
Energy Balance climate model. The CG will be coupled to the
Glacial Systems Model (GSM) for
experiments over the last glacial cycle. We expect through the
development of our CG, the GSM
will be provided with enhanced climate forcing (Temperature and
precipitation) relative to previous
46
-
experiments. To simulate more atmospheric variables (like
evaporation, etc.) the CG needs to be
retrained with those GCM fields. In future work, the CG will
tested for use with all the major
ice-sheets of the last glacial cycle. The CG approach will also
be implemented with more advanced
EMICs (e.g. LOVECLIM) for shorter time scale contexts (given
their increased computational
expense).
2.9 Co-authorship statement
M. Arif wrote the initial draft of this manuscript and carried
out much of the implementation and
testing. T. Hauser started the project implementation. L.
Tarasov created the “Climate Generator”
and “Climate Turing Test” concepts, oversaw research design, and
heavily edited this manuscript.
47
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